Dynamic behaviors of a predator–prey model perturbed by a complex type of noises

Asymptotic behaviour of a non‐autonomous multispecies Holling type II model with a complex type of noises

“…Our equation differs from the aforementioned equation (A) examined in (Guo et al, 2019). To investigate the extinction property of our method, we will start by considering the equation on the boundary, $$$$ \tilde{x}(t)=\tilde{x}(t)\left\{{r}^u+{a}^u\tilde{x}(t)\right\}+{\sigma}^u\tilde{x}(t) dB(t), $$$$ with initial value $$$ \tilde{x}(0)=x(0)>0 $$$ almost surely, following the lemma 2.2* and solving the Fokker–Planck equation in (Guo et al, 2019) we have the density of the stationary distribution of the process $$$ \tilde{x}(t) $$$, $$$$ \phi (x)=\frac{l^q}{\Gamma (q)}{x}^{q-1}{e}^{- lx},x>0, $$$$ where $$$ l=2{a}^u/{\left({\sigma}^u\right)}^2 $$$, $$$ q=2{r}^u/{\left({\sigma}^u\right)}^2-1>0 $$$ and $$$ \Gamma \left(\cdot \right) $$$ is the Gamma function. By strong ergodicity theorem, we can deduce that for any measurabl...…”

“…To establish the extinction property, it is essential to introduce certain preliminaries. Let us begin by restating the lemma 2.2 in (Guo et al, 2019),…”

“…Lemma 2.2* Guo et al, 2019. There exists a unique conditions positive solution e x t ð Þ to SDE (A) for any x 0 ð Þ> 0, which is global and represented by…”

“…To establish the extinction property, it is essential to introduce certain preliminaries. Let us begin by restating the lemma 2.2 in (Guo et al, 2019), where they considered the following equation (A) on the boundary, $$$$ d\tilde{x}(t)=\tilde{x}(t)\left(r-b\tilde{x}(t)\right) dt+\sigma \tilde{x}(t) dB(t)\kern6em \left(\mathrm{A}\right) $$$$ with initial $$$ \tilde{x}(0)=x(0)>0 $$$, $$$ \forall t\ge 0 $$$ almost surely.Lemma There exists a unique conditions positive solution $$$ \tilde{x}(t) $$$ to SDE (A) for any $$$ x(0)>0 $$$, which is global and represented by $$$$ \tilde{x}(t)=\frac{\exp \left\{{\int}_0^t\left(r-\frac{1}{2}{\sigma}^2\right) ds+\sigma dB(s)\right\}}{\frac{1}{x(0)}+{\int}_0^tb\cdot \exp \left\{{\int}_0^s\left(r-\frac{1}{2}{\sigma}^2\right) du+\sigma dB(u)\right\} ds}. $$$$…”

“…However, most of these models concentrate on introducing stochasticity through linear environmental noise affecting the populations or the functional response, failing to account for the diverse impact of different noises on the dynamical system. To address this limitation, a pioneering approach was introduced, expanding the type of noises from simple linear perturbations to non-linear types in two autonomous models and a non-autonomous model, namely the stochastic SIR model with Beddington-DeAngelis functional response (Du & Nhu, 2017), the predator-prey model with Crowly-Martin type functional response (Guo et al, 2019) and the Holling type II predator-prey model (Wei & Li, 2022). In the current literature, there is a notable scarcity of discussions regarding stochastic non-autonomous multispecies Holling type II models with simple linear noise, let alone the exploration of stochastic versions with complex noise patterns.…”